|Mar 10, 2023
The SYZ conjecture asserts that for a Calabi-Yau manifold, the mirror pair should carry dual special Lagrangian torus fibrations. Extending beyond the Calabi-Yau case, to semi-Fano Log Calabi-Yau surfaces to be precise, one should consider what is known as the Landau-Ginzburg model, i.e. a Laurent polynomial defined on the mirror space (which is called the potential). Using new tropical geometric ideas, we give a closed-string version of the mirror symmetry for such semi-Fano Log Calabi-Yau surfaces, that is, we show isomorphism between the quantum cohomology of the surface and the Jacobian ideal ring of the potential.